Morphometrics: variation in shape and its covariation with other variables
Position and orientation are due to digitizing procedures, and are not biologically relevant for most applications
Scale includes the combined effects of focal distance during digitizing and “real” size variation
Digitizing scale is calibrated during data acquisition
Size is of biological interest. So, we standardize for it to obtain shape variables, but record it for subsequent analyses
Small objects: landmarks are closer together
Large objects: landmarks are further apart
So, in GM, size is associated to the dispersion of landmark coordinates
Centroid size: the square root of the sum of the squared distances between each landmark and the centroid (center of mass of the object) of the landmark configuration:
\[\small{CS}=\sqrt{\sum_{i,j}^{k,p}\left(\mathbf{Y}_{ij}-\mathbf{Y}_{ic}\right)^2}\]
where \(\small{p}\) is the number of landmarks and \(\small{k}\) is the number of coordinate dimensions
\[\small{D}_{Proc}=\sqrt{\sum_{i,j}^{k,p}\left(\mathbf{Y}_{1.ij}-\mathbf{Y}_{2.ij}\right)^2}\]
\(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)
\[\small{CS}=\sqrt{\sum_{i,j}^{k,p}\left(\mathbf{Y}_{ij}-\mathbf{Y}_{ic}\right)^2}=\]
\[\small{[tr}[\mathbf{(Y-\overline{Y})(Y-\overline{Y})^T}]]^{-1/2}=1\]
\(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)
Perform OPA in iterative fashion
Sexual dimorphism in head shape of Podarcis wall lizards
12 lateral 2D landmarks
Males and females
MANOVA to test for shape differences
PCA to visualize shape space
## Df SS MS Rsq F Z Pr(>F)
## gp 1 0.009180 0.0091801 0.09059 5.8772 4.5952 0.001 **
## Residuals 59 0.092158 0.0015620 0.90941
## Total 60 0.101338
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## [1] 0.019120833 0.014962558 0.013953399 0.011872076 0.011252644
## [6] 0.010428595 0.010172139 0.008714636 0.008158964 0.007586907
## [11] 0.006831716 0.006049213 0.005992073 0.005275534 0.004900721
## [16] 0.004129789 0.003339428 0.003200212 0.002954954 0.002591265
## [21] 0.000000000 0.000000000 0.000000000 0.000000000
Now we have: \(\tiny{Y}=\begin{bmatrix} X_1 & Y_1 & Z_1 \\ X_2 & Y_2 & Z_2 \\ \vdots & \vdots & \vdots \\ X_p & Y_p & Z_p \\ \end{bmatrix}\) & \(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)
Now we have: \(\tiny{Y}=\begin{bmatrix} X_1 & Y_1 & Z_1 \\ X_2 & Y_2 & Z_2 \\ \vdots & \vdots & \vdots \\ X_p & Y_p & Z_p \\ \end{bmatrix}\) & \(\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\)
If shape variation is localized in a few landmarks this can be a problem - The ‘Pinocchio effect’
One may desire methods resistant to this effect
Estimate parameters for translation, scaling and rotation using medians instead of least-squares means
Example: two shapes that differ only in the front triangle
\[\tiny{D}_{Proc}=\sqrt{\sum_{i,j}^{k,p}\left(\mathbf{Y}_{1.ij}-\mathbf{Y}_{2.ij}\right)^2}\]
